Manuscript X-K-ST@1 · X-K-ST.tex · corollary · 93c39c0f

Corollary: Several supplied clocks without intervening coupling

The multi-instrument composition law › An exact many-clock law · lines 865–889 · proof included or linked

X-K-ST@1#cor:manyclock

Let $C_{1}, \dots, C_{m}, S$ be supplied jointly at the initial cut, with no independence assumption. For successive destructive readout in the specified bases and no intervening coupling, the complete channel is

X ⟼ k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†}, D_{k} = ⟨ τ^{(1)}_{k_{1}} ∣ \dots ⟨ τ^{(m)}_{k_{m}} ∣ \otimes I_{S} .

Discarding every record gives $Tr_{C_{1} \dots C_{m}} X$ , including all references.

Proof. Apply Theorem thm:oneclock to different live factors in succession. The resulting Kraus bra is $D_{k}$ . Summing its diagonal blocks gives the joint partial trace.

Displayed formulas in this passage

unlabeled · line 871
$X ⟼ k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†}, D_{k} = ⟨ τ^{(1)}_{k_{1}} ∣ \dots ⟨ τ^{(m)}_{k_{m}} ∣ \otimes I_{S} .$

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section: An exact many-clock law